Where did you get the factor of 3/5 from? There is no [itex]\alpha[/itex] = 3/5 in gravitational potential

The gravitational potential energy is:
[itex]\frac{-GMm}{r^2}[/itex]

The gravitational binding energy can be thought of as an internal potential energy. How much grav potential is stored in an object? You must compute the gravitational potential for every particle in the object relative to all the rest of them which simply yields

Where did you get the factor of 3/5 from? There is no [itex]\alpha[/itex] = 3/5 in gravitational potential

The gravitational potential energy is:
[itex]\frac{-GMm}{r^2}[/itex]

Was this a typo? The gravitational force varies as [itex]1/r^2[/itex] but the gravitational potential energy as 1/r, the integral of [itex]1/r^2[/itex].

The gravitational binding energy can be thought of as an internal potential energy. How much grav potential is stored in an object? You must compute the gravitational potential for every particle in the object relative to all the rest of them which simply yields

Sorry, I was confused at to what the question was. Were you asking if your equation was correct? Because yes, it is.

Except, it's not equal to the NEGATIVE of the total gravitational potential energy, because gravitational potential is also negative. Both quantities are negative, it's just the gravitational binding the the total internal gravitational potential of the object.

Staff: Mentor

Binding energy is the energy that we must add to a bound system in order to separate it completely into its constituents. It is always positive. If it were negative, the system could fly apart by itself, releasing energy. For a gravitationally bound sphere with uniform density, it is Eb = 3GM2/5R.

When we add energy to a system, we increase its (potential + kinetic) energy. If the kinetic energy of the "pieces" is zero before and after, then only the potential energy changes. Ubound + Eb = Useparated. In the situation that we're dealing with here, we define the potential energy to be zero when the system is completely separated, that is, Useparated = 0.

I think what jtbell wrote is useful. Often, an object in a bound state, like an atom in a ground state, or a spherical planet, must be at a minimum of potential energy.

The easiest way to do this is to say that a potential energy of zero means the system is totally unbound (imagine an ionized atom), and obviously, the bound state would have to have a negative energy, -3GM2/5R, for a planet, star, etc.

You can also think of the positive of this as being the energy required to unbind the system, so the energy needed to ionize an atom, or the energy required by the Death Star to disintegrate Aldreraan ;)

There is an arbirarity to potential energy. You can set the zero point energy to be anything as long as you are consistent. There are standard conventions for some systems like gravitational binding energy that make everything much nicer, however.

I see that they use the positive and negative version of the equation. It makes little difference. It only matters the convention. The top equation is essentially saying that this is the energy stored in the gravitational binding of the object, and the second equation is saying that this is the total binding potential of the spherical object , which is negative if we say that the potential zero point is when the system is completely unbound.